Theorem rspc2vd 3196

Description: Deduction version of 2-variable restricted specialization, using implicit substitution. Notice that the class D for the second set variable y may depend on the first set variable x. (Contributed by AV, 29-Mar-2021.)

Hypotheses

Ref	Expression
rspc2vd.a	|- ( x = A -> ( th <-> ch ) )
rspc2vd.b	|- ( y = B -> ( ch <-> ps ) )
rspc2vd.c	|- ( ph -> A e. C )
rspc2vd.d	|- ( ( ph /\ x = A ) -> D = E )
rspc2vd.e	|- ( ph -> B e. E )

Assertion

Ref	Expression
rspc2vd	|- ( ph -> ( A. x e. C A. y e. D th -> ps ) )

Distinct variable groups:   x, A, y   y, B   x, C   y, D   x, E   ph, x   ch, x   ps, y

Allowed substitution hints:   ph( y)   ps( x)   ch( y)   th( x, y)   B( x)   C( y)   D( x)   E( y)

Proof of Theorem rspc2vd

Step	Hyp	Ref	Expression
1		rspc2vd.e	. . 3 |- ( ph -> B e. E )
2		rspc2vd.c	. . . 4 |- ( ph -> A e. C )
3		rspc2vd.d	. . . 4 |- ( ( ph /\ x = A ) -> D = E )
4	2, 3	csbied 3174	. . 3 |- ( ph -> [_ A / x ]_ D = E )
5	1, 4	eleqtrrd 2311	. 2 |- ( ph -> B e. [_ A / x ]_ D )
6		nfcsb1v 3160	. . . . 5 |- F/_ x [_ A / x ]_ D
7		nfv 1576	. . . . 5 |- F/ x ch
8	6, 7	nfralw 2569	. . . 4 |- F/ x A. y e. [_ A / x ]_ D ch
9		csbeq1a 3136	. . . . 5 |- ( x = A -> D = [_ A / x ]_ D )
10		rspc2vd.a	. . . . 5 |- ( x = A -> ( th <-> ch ) )
11	9, 10	raleqbidv 2746	. . . 4 |- ( x = A -> ( A. y e. D th <-> A. y e. [_ A / x ]_ D ch ) )
12	8, 11	rspc 2904	. . 3 |- ( A e. C -> ( A. x e. C A. y e. D th -> A. y e. [_ A / x ]_ D ch ) )
13	2, 12	syl 14	. 2 |- ( ph -> ( A. x e. C A. y e. D th -> A. y e. [_ A / x ]_ D ch ) )
14		rspc2vd.b	. . 3 |- ( y = B -> ( ch <-> ps ) )
15	14	rspcv 2906	. 2 |- ( B e. [_ A / x ]_ D -> ( A. y e. [_ A / x ]_ D ch -> ps ) )
16	5, 13, 15	sylsyld 58	1 |- ( ph -> ( A. x e. C A. y e. D th -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   [_csb 3127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by:  insubm  13567